Characterizations of rectifiable metric measure spaces

Abstract

We characterize $n$-rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite $n$-densities and one of the following: is an $n$-dimensional Lipschitz differentiability space; has $n$-independent Alberti representations; satisfies David's condition for an $n$-dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisfies David's condition. This allows us to apply previously known "biLipschitz pieces" results on the charts.

35 pages. Provided nine additional pages of details and mentioned a general biLipschitz decomposition theorem. Updated introduction to be more self-contained. No significant technical changes